Neumann vs Hankel function as solutions and why computer programs are not the same

It is well known that the solution of Bessel differential equation can be: 1) Bessel first kind, J, and Neumann, Y, (Bessel 2nd kind) or 2) two Hankel functions or 3) Bessel 1st kind, J, and Hankel, H (not commonly used ) [login to view URL] In the attached paper, they used option 1: i.e. had an equation B*J+C*Y if we use B*J+C*H values of B and C should change so the equation would remain since H=J+i*Y but when I tried to change to option 3. Results are not the same! why should not they be the same? did I do a mistake? if there is a mistake where?